1.12 Confirming Continuity over an Interval

• Match the condition for continuity with its requirement:

  Limit Exists ↔️ \lim_{x \rightarrow c} f(x) \text{ \exists}

  Function Value Exists ↔️ f(c) \text{ \exists}

  Limit Equals Function Value ↔️ $\lim_{x \rightarrow c} f(x) =$$f(c)$
• A function is continuous at a point c if the limit of f(x) as x approaches c exists.


  True
• A function f(x) is continuous over a closed interval [a, b] if it is continuous at every point within the open interval (a, b).


  True
• Is f(x) = √(1 - x^2) continuous over the closed interval [-1, 1]?

  Yes
• Right continuity at a requires that \lim_{x \to a^+} f(x) = f(a)</latex>


  True
• A function f(x) is continuous over an open interval (a, b) if lim x→c f(x) = f(c) for every c ∈ (a, b)


  True
• Match the condition for continuity with its requirement:

  Limit Exists ↔️ $\lim_{x \to c} f(x)$ exists

  Function Value Exists ↔️ f(c) exists

  Limit Equals Function Value ↔️ $\lim_{x \to c} f(x) =$$f(c)$
• Steps to determine continuity over a closed interval [a, b]

  1️⃣ Check continuity at every point within (a, b)

  2️⃣ Verify right continuity at a

  3️⃣ Verify left continuity at b
• A jump discontinuity occurs when the limit of the function does not exist, and the function value at that point does not exist either.


  True
• A removable discontinuity occurs when the limit exists, but the function value at that point does not exist.

  True
• An infinite discontinuity occurs when the function "blows up" or approaches positive or negative infinity.


  True
• A removable discontinuity can be fixed by redefining the function at the point where it occurs.


  True
• The function f(x) = (x^2 - 4) / (x - 2) has an infinite discontinuity at x = 0 because the limit approaches ±∞.


  True
• Why is a non-removable discontinuity called "non-removable"?

  The limit does not exist
• What is the Intermediate Value Theorem (IVT)?

  Ensures a value within an interval
• What is the first step in verifying if the IVT applies to a function?

  Check for continuity
• What is the conclusion of the example when applying the IVT to f(x) = x^2 - 3x + 1 on [1, 4]?

  There exists a value c ≈ 3.303
• What are the three conditions for a function to be continuous over an interval [a, b]?

  Limit exists, value exists, limit equals value
• The limit of f(x) as x approaches c must exist for f(x) to be continuous at c.


  True
• For a function f(x) to be continuous at a point c, the limit of f(x) as x approaches c must exist
• For continuity over a closed interval [a, b], the limit of f(x) as x approaches a from the right must equal f(a)
• A function f(x) is continuous over a closed interval [a, b] if it satisfies the following conditions: continuity at each point, right continuity at a, and left continuity at b
• Consider the function f(x) = √(1 - x^2) defined over [-1, 1]. It satisfies continuity at every point within the interval (-1, 1)
• What is the limit of f(x) = 1/x as x approaches c within (0, ∞)?

  \lim_{x \to c} \frac{1}{x} = \frac{1}{c}</latex>
• The limit of f(x) as x approaches c must exist for continuity.

  True
• What is the requirement for right continuity at the left endpoint a of a closed interval [a, b]?

  $\lim_{x \to a^ + } f(x) =$$f(a)$
• Match the type of discontinuity with its requirement:

  Jump Discontinuity ↔️ Limit does not exist, f(c) does not exist

  Removable Discontinuity ↔️ Limit exists, f(c) does not exist

  Infinite Discontinuity ↔️ Limit does not exist, f(c) exists
• A removable discontinuity can be fixed by redefining the function at the point
• The function f(x) = (x^2 - 4) / (x - 2) has a removable discontinuity at x = 2
• An infinite discontinuity occurs when the function approaches positive or negative infinity
• Match the type of discontinuity with its solution:

  Removable Discontinuity ↔️ Redefine the function at c

  Non-Removable Discontinuity ↔️ Cannot be fixed without changing behavior
• The function f(x) = (x² - 4) / (x - 2) has a removable discontinuity at x = 2 because lim x→2 f(x) = 4.


  True
• The IVT requires that the function f(x) must be continuous on a closed interval [a, b].


  True
• The interval [1, 4] is an example of a closed interval.


  True
• For a function to be continuous at a point c, the limit of f(x) as x approaches c must equal the function value f(c)
• For continuity at a point c, the limit of f(x) as x approaches c must equal the function value f(c)
• A function f(x) is continuous at a point c if the function value f(c) must exist
• Arrange the conditions for continuity over a closed interval [a, b] in the correct order:

  1️⃣ Continuity at Each Point: $\lim_{x \rightarrow c} f(x) =$$f(c)$ for all $c \in (a, b)$

  2️⃣ Right Continuity at a: $\lim_{x \rightarrow a^{ + }} f(x) =$$f(a)$

  3️⃣ Left Continuity at b: $\lim_{x \rightarrow b^{ - }} f(x) =$$f(b)$
• What is the requirement for continuity at each point within an open interval (a, b)?

  $\lim_{x \to c} f(x) =$$f(c)$
• Is f(x) = √(1 - x^2) continuous over the closed interval [-1, 1]?

  Yes